By choosing some special (random) initial data, we prove that with probability $1$, the stochastic shadow Gierer-Meinhardt system blows up pointwisely in finite time.We also give a (random) upper bound for the blowup time and some estimates about this bound. By increasing the amplitude of the initial data, we can get the blowup in any short time with positive probability.
References
Chow, Pao-Liu. Nonlinear stochastic wave equations: blow-up of second moments in $L^ 2$-norm. Ann. Appl. Probab. 19 (2009), no. 6, 2039–2046. MR2588238 10.1214/09-AAP602 euclid.aoap/1259158765
Chow, Pao-Liu. Nonlinear stochastic wave equations: blow-up of second moments in $L^ 2$-norm. Ann. Appl. Probab. 19 (2009), no. 6, 2039–2046. MR2588238 10.1214/09-AAP602 euclid.aoap/1259158765
Chow, Pao-Liu. Explosive solutions of stochastic reaction-diffusion equations in mean $L^ p$-norm. J. Differential Equations 250 (2011), no. 5, 2567–2580. MR2756076 10.1016/j.jde.2010.11.008Chow, Pao-Liu. Explosive solutions of stochastic reaction-diffusion equations in mean $L^ p$-norm. J. Differential Equations 250 (2011), no. 5, 2567–2580. MR2756076 10.1016/j.jde.2010.11.008
Chow, Pao-Liu; Khasminskii, Rafail. Almost sure explosion of solutions to stochastic differential equations. Stochastic Process. Appl. 124 (2014), no. 1, 639–645. MR3131308 10.1016/j.spa.2013.09.006Chow, Pao-Liu; Khasminskii, Rafail. Almost sure explosion of solutions to stochastic differential equations. Stochastic Process. Appl. 124 (2014), no. 1, 639–645. MR3131308 10.1016/j.spa.2013.09.006
Chow, Pao-Liu; Liu, Kai. Positivity and explosion in mean $L^ p$-norm of stochastic functional parabolic equations of retarded type. Stochastic Process. Appl. 122 (2012), no. 4, 1709–1729. MR2914769 10.1016/j.spa.2012.01.012Chow, Pao-Liu; Liu, Kai. Positivity and explosion in mean $L^ p$-norm of stochastic functional parabolic equations of retarded type. Stochastic Process. Appl. 122 (2012), no. 4, 1709–1729. MR2914769 10.1016/j.spa.2012.01.012
Friedman, Avner; McLeod, Bryce. Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J. 34 (1985), no. 2, 425–447. MR783924 10.1512/iumj.1985.34.34025Friedman, Avner; McLeod, Bryce. Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J. 34 (1985), no. 2, 425–447. MR783924 10.1512/iumj.1985.34.34025
Hu, Bei; Yin, Hong-Ming. Semilinear parabolic equations with prescribed energy. Rend. Circ. Mat. Palermo (2) 44 (1995), no. 3, 479–505. MR1388759 10.1007/BF02844682Hu, Bei; Yin, Hong-Ming. Semilinear parabolic equations with prescribed energy. Rend. Circ. Mat. Palermo (2) 44 (1995), no. 3, 479–505. MR1388759 10.1007/BF02844682
Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8 MR1121940Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8 MR1121940
Kelkel, Jan; Surulescu, Christina. On a stochastic reaction-diffusion system modeling pattern formation on seashells. J. Math. Biol. 60 (2010), no. 6, 765–796. MR2606514 10.1007/s00285-009-0284-5Kelkel, Jan; Surulescu, Christina. On a stochastic reaction-diffusion system modeling pattern formation on seashells. J. Math. Biol. 60 (2010), no. 6, 765–796. MR2606514 10.1007/s00285-009-0284-5
Li, Fang; Ni, Wei-Ming. On the global existence and finite time blow-up of shadow systems. J. Differential Equations 247 (2009), no. 6, 1762–1776. MR2553858 10.1016/j.jde.2009.04.009Li, Fang; Ni, Wei-Ming. On the global existence and finite time blow-up of shadow systems. J. Differential Equations 247 (2009), no. 6, 1762–1776. MR2553858 10.1016/j.jde.2009.04.009
Li, Fang; Yip, Nung Kwan. Finite time blow-up of parabolic systems with nonlocal terms. Indiana Univ. Math. J. 63 (2014), no. 3, 783–829. MR3254524 10.1512/iumj.2014.63.5253Li, Fang; Yip, Nung Kwan. Finite time blow-up of parabolic systems with nonlocal terms. Indiana Univ. Math. J. 63 (2014), no. 3, 783–829. MR3254524 10.1512/iumj.2014.63.5253
Lieberman, Gary M. Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. xii+439 pp. ISBN: 981-02-2883-X MR1465184Lieberman, Gary M. Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. xii+439 pp. ISBN: 981-02-2883-X MR1465184
Mueller, Carl. Singular initial conditions for the heat equation with a noise term. Ann. Probab. 24 (1996), no. 1, 377–398. MR1387640 10.1214/aop/1042644721 euclid.aop/1042644721
Mueller, Carl. Singular initial conditions for the heat equation with a noise term. Ann. Probab. 24 (1996), no. 1, 377–398. MR1387640 10.1214/aop/1042644721 euclid.aop/1042644721
Mueller, Carl; Sowers, Richard. Blowup for the heat equation with a noise term. Probab. Theory Related Fields 97 (1993), no. 3, 287–320. MR1245247 10.1007/BF01195068Mueller, Carl; Sowers, Richard. Blowup for the heat equation with a noise term. Probab. Theory Related Fields 97 (1993), no. 3, 287–320. MR1245247 10.1007/BF01195068
Ni, Wei-Ming; Suzuki, Kanako; Takagi, Izumi. The dynamics of a kinetic activator-inhibitor system. J. Differential Equations 229 (2006), no. 2, 426–465. MR2263562 10.1016/j.jde.2006.03.011Ni, Wei-Ming; Suzuki, Kanako; Takagi, Izumi. The dynamics of a kinetic activator-inhibitor system. J. Differential Equations 229 (2006), no. 2, 426–465. MR2263562 10.1016/j.jde.2006.03.011
Wei, Juncheng; Winter, Matthias. Mathematical aspects of pattern formation in biological systems. Applied Mathematical Sciences, 189. Springer, London, 2014. xii+319 pp. ISBN: 978-1-4471-5525-6; 978-1-4471-5526-3 MR3114654Wei, Juncheng; Winter, Matthias. Mathematical aspects of pattern formation in biological systems. Applied Mathematical Sciences, 189. Springer, London, 2014. xii+319 pp. ISBN: 978-1-4471-5525-6; 978-1-4471-5526-3 MR3114654
M. Winter, L. Xu, J. Zhai and T. Zhang: The dynamics of the stochastic shadow Gierer-Meinhardt System, J. of Differential Equations (to appear). MR3411665 10.1016/j.jde.2015.08.047M. Winter, L. Xu, J. Zhai and T. Zhang: The dynamics of the stochastic shadow Gierer-Meinhardt System, J. of Differential Equations (to appear). MR3411665 10.1016/j.jde.2015.08.047
